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One of the basic theorems in the general theory of dynamical systems with an [[Invariant measure|invariant measure]] (cf. also [[Ergodic theory|Ergodic theory]]).

Let the motion of a system be described by the differential equations

$$ \tag{1 }

\frac{d x _ {i} }{dt}
  = \ 
X _ {i} ( x _ {1} \dots x _ {n} ) ,\  i = 1 \dots n ,
$$

where the single-valued functions  $  X _ {i} ( x _ {1} \dots x _ {n} ) $
satisfy the condition

$$ 
\sum _ { i= } 1 ^ { n } 

\frac{\partial  ( M X _ {i} ) }{\partial  x _ {i} }
  =  0 ,\ \ 
M  >  0 ,
$$

so that equations (1) admit a positive [[Integral invariant|integral invariant]]

$$ \tag{2 }
\int\limits _ { V } M  d x _ {1} \dots d x _ {n} .
$$

It is also assumed that if there exists a certain domain  $  V $
of finite volume such that if a moving point  $  P $
with coordinates  $  x _ {1} \dots x _ {n} $
is found inside  $  V $
at the initial moment of time  $  t _ {0} $, 
then it will remain inside this domain for an arbitrary long time and

$$ 
\int\limits _ { V } M  d x _ {1} \dots d x _ {n}  <  \infty .
$$

The Poincaré return theorem: If one considers a domain  $  U _ {0} $
contained in  $  V $, 
then there is an infinite choice of initial positions of the point  $  P $
such that the trajectory of  $  P $
intersects the domain  $  U _ {0} $
an infinite number of times. If this choice of the initial position is made at random inside  $  U _ {0} $, 
then the probability that the point  $  P $
does not intersect the domain  $  U _ {0} $
an infinite number of times will be infinitely small.

In other words, if the initial conditions are not exceptional in the sense indicated, then the point  $  P $
passes infinitely often arbitrarily near to its initial position.

H. Poincaré called a motion in which the system returns an infinite number of times to a neighbourhood of the initial state stable in the sense of Poisson (see [[Poisson stability|Poisson stability]]). The Poincaré return theorem was first established by Poincaré (see [[#References|[1]]] and [[#References|[2]]]) and its proof was improved by C. Carathéodory [[#References|[3]]].

Carathéodory used four axioms to introduce the abstract concept of the measure  $  \mu A $
of any set  $  A \subset  R $
of a metric space  $  R $, 
and considered a dynamical system  $  f ( p , t ) $(
$  p = P $
for  $  t = 0 $) 
in  $  R $; 
he then called the measure invariant with respect to the system  $  f ( p , t ) $
if for any  $  \mu $-
measurable set  $  A $,

$$ 
\mu f ( A , t )  =  \mu A ,\  - \infty < t < + \infty .
$$

An invariant measure is the natural generalization of the integral invariants (2) for the differential equations (1). Assuming the measure of the whole space  $  R $
to be finite, Carathéodory proved that:

1) if  $  \mu A = m > 0 $, 
then values  $  t $
can be found,  $  | t | \geq  1 $, 
such that  $  \mu [ A \cdot f ( A , t ) ] > 0 $, 
where  $  A \cdot f ( A , t ) $
is the set of points belonging simultaneously to the sets  $  A $
and  $  f ( A , t ) $;

2) if in a space  $  R $
with a countable base,  $  \mu R = 1 $
for the invariant measure  $  \mu $, 
then almost-all points  $  p \in R $(
in the sense of the measure  $  \mu $) 
are stable in the sense of Poisson.

A.Ya. Khinchin [[#References|[5]]] made part 1) of this theorem more precise by proving that for each measurable set  $  E $, 
$  \mu E = m > 0 $, 
and for any  $  t $, 
$  - \infty < t < + \infty $, 
the inequality

$$ 
\mu ( t)  =  \mu ( E \cdot f ( E , t ) )  >  \lambda m  ^ {2}
$$

is satisfied for a relatively-dense set of values of  $  t $
on the axis  $  - \infty < t < + \infty $(
for any  $  \lambda < 1 $).

N.G. Chetaev (see [[#References|[6]]], [[#References|[7]]]) generalized Poincaré's theorem for the case when the functions  $  X _ {i} $
in (1) depend also periodically on the time  $  t $. 
Namely, let a) only real values of variables correspond to the real states of the system; b) the functions  $  X _ {i} $
in the differential equations (1) of the motion be periodic with respect to  $  t $
with a single period  $  \tau $
common to them all; c) throughout its motion, the point  $  P $
does not leave a certain closed domain  $  R $
if its initial position  $  P _ {0} $
is somewhere inside a given domain  $  W _ {0} $; 
d)  $   \mathop{\rm mes}  W _ {k} \geq  a  \mathop{\rm mes}  W _ {0} $, 
where  $   \mathop{\rm mes}  W _ {k} = \int _ {W _ {k}  } d x _ {1} \dots d x _ {n} $
denotes the measure of the set  $  W _ {k} $(
volume in the sense of Lebesgue) which consists of those moving points at time  $  t = t _ {0} + k \tau $
which started at time  $  t _ {0} $
from  $  W _ {0} $; 
$  k $
is a certain integer, and it is assumed that the constant  $  a $
is not infinitesimally small. Then almost-everywhere in the domain  $  W _ {0} $(
apart perhaps on a set of measure zero) the trajectories are stable in the sense of Poisson.

N.M. Krylov and N.N. Bogolyubov [[#References|[8]]] described the structure of the invariant measure with respect to the given dynamical system for a very wide class of dynamical systems (see also [[#References|[4]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,   "Sur le problème des trois corps et les équations de la dynamique"  ''Acta. Math.'' , '''13'''  (1890)  pp. 1–270</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Poincaré,   "Sur le problème des trois corps et les équations de la dynamique" , ''Oeuvres'' , '''XII''' , Gauthier-Villars  (1952)  pp. 262–479 (in particular, p. 314)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Carathéodory,   "Ueber den Wiederkehrsatz von Poincaré"  ''Sitz. Ber. Preuss. Akad. Wiss. Berlin''  (1919)  pp. 580–584</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.Ya. Khinchin,   "Eine Verschärfung des Poincaréschen Wiederkehrsatzes"  ''Comp. Math.'' , '''1'''  (1934)  pp. 177–179</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.G. Chetaev,   "Sur la stabilité à la Poisson"  ''C.R. Acad. Sci. Paris'' , '''187'''  (1928)  pp. 637–638</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N.G. Chetaev,   ''Uchen. Zap. Kazan. Univ.'' , '''89''' :  2  (1929)  pp. 199–201</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  N.N. Krylov,   N.N. Bogolyubov,   "La théorie de la mesure dans son application à l'etude des systèmes dynamiques de la mécanique non-linéaire"  ''Ann. of Math.'' , '''38''' :  1  (1937)  pp. 65–113</TD></TR></table>

====Comments====
In the literature the result discussed above is also often called the Poincaré recurrence theorem.

The set  $  U _ {0} $
in the theorem need not be open: the theorem is true provided only that  $  \mu ( U _ {0} ) > 0 $. 
The recurrence theorem is valid for volume-preserving flows on Riemannian manifolds  $  V $
of finite volume. The recurrence theorem is also true for a discrete-time dynamical system, e.g. for a mapping  $  f $
of a bounded domain in Euclidean space to itself that preserves Lebesgue measure. See [[#References|[a1]]] for another generalization.

There seems to be an incompatibility of the prediction by Poincaré's recurrence theorem (namely, almost surely a system will recur arbitrarily close to its original state) with the conclusions of thermodynamics as the second law and the [[Boltzmann H-theorem|Boltzmann  $  H $-
theorem]] (increasing entropy). In this respect the following estimation of the expected recurrence time is of interest: it is  $  1/ \mu ( E) $, 
where  $  E $
denotes the  "event"  that recurs  $  ( \mu ( E) > 0) $; 
for practical situations this time is much larger than the lifetime of the universe (by factors like  $  2  ^ {100} $); 
see [[#References|[a2]]].

The Poincaré recurrence theorem was used by S. Kakutani as the basis for an important construction: that of the induced or derivative transformation of a [[Measure-preserving transformation|measure-preserving transformation]] (with as a reverse construction that of a primitive transformation). See [[#References|[a3]]] or [[#References|[a4]]], pp. 39, 40.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.R. Halmos,   "Invariant measures"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 735–754</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Kac,   "On the notion of recurrence in discrete stochastic processes"  ''Bull. Amer. Math. Soc.'' , '''53'''  (1947)  pp. 1002–1010</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Kakutani,   "Induced measure preserving transformations"  ''Proc. Japan. Acad.'' , '''19'''  (1943)  pp. 635–641</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Petersen,   "Ergodic theory" , Cambridge Univ. Press  (1983)  pp. 39</TD></TR></table>